The Loewner differential equation has led to inequalities for univalent functions that played an important role in the solution of the Bieberbach conjecture by Louis de Branges in 1985.
22.
Inequalities for univalent functions on the unit disk can be proved by using the density for uniform convergence on compact subsets of "'slit mappings " '.
23.
A fundamental characteristic of the univalent foundations is that they, when combined with the Martin-L�f type theory, provide a practical system for formalization of modern mathematics.
24.
Most of the work on formalization of mathematics in the framework of univalent foundations is being done using various sub-systems and extensions of the Calculus of Inductive Constructions.
25.
Originally, the univalent foundations were devised by Vladimir Voevodsky with the goal of enabling those who work in classical pure mathematics to use computers to verify their theorems and constructions.
26.
These features, called univalent or privative features, can only describe the classes of segments that are said to possess those features, and not the classes that are without them.
27.
A more detailed discussion of the history of some of the ideas that contribute to the current state of the univalent foundations can be found at the page on homotopy type theory.
28.
An account of the main ideas of the univalent foundations and their connection to constructive mathematics can be found in a tutorial by Thierry Coquand ( part 1, part 2 ).
29.
A Tutton salt is a double sulfate of the typical formula, where " A " is a univalent cation, and " B " a divalent metal ion.
30.
A considerable amount of mathematics has been formalized using this system and modern proof assistants such as HoTT Coq library and HoTT Agda library, that developed the univalent ideas in new directions.
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